On the Shannon entropy of the number of vertices with zero in-degree in randomly oriented hypergraphs

TL;DR

This paper investigates the maximum Shannon entropy of the number of distinct colours observed after rolling multiple dice with coloured sides, establishing an upper bound and conjecturing optimal colour distribution.

Contribution

It provides an upper bound on the entropy of the observed colours and employs novel entropy bounds and binomial entropy analysis to support the conjecture on optimal colouring.

Findings

Entropy of observed colours is at most (1/2) log(n) + O(1)

Bound is tight for equal numbers of coins and colours

Entropy maximization conjecture for even colour distribution

Abstract

Suppose that you have $n$ colours and $m$ mutually independent dice, each of which has $r$ sides. Each dice lands on any of its sides with equal probability. You may colour the sides of each die in any way you wish, but there is one restriction: you are not allowed to use the same colour more than once on the sides of a die. Any other colouring is allowed. Let $X$ be the number of different colours that you see after rolling the dice. How should you colour the sides of the dice in order to maximize the Shannon entropy of $X$? In this article we investigate this question. We show that the entropy of $X$ is at most $\frac{1}{2} \log(n) + O(1)$ and that the bound is tight, up to a constant additive factor, in the case of there being equally many coins and colours. Our proof employs the differential entropy bound on discrete entropy, along with a lower bound on the entropy of binomial random variables whose outcome is conditioned to be an even integer. We conjecture that the entropy is maximized when the colours are distributed over the sides of the dice as evenly as possible.